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# Questions tagged [plane-geometry]

Plane Geometry is about flat shapes like lines, circles and triangles , shapes that can be drawn on a piece of paper

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### Recovering a set from its projections in varying coordinate systems - a projection hull?

Let me describe the simplest non-trivial case of what I have in mind. Let $V$ be a 2-dimensional $\mathbb{R}$-vector space and fix an isomorphism $V \cong \mathbb{R}^2$, where $\mathbb{R}^2$ is ...
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### Is every weakly $1$-dimensional space embeddable in the plane?

A $1$-dimensional (separable metric) space $X$ is weakly $1$-dimensional if $$\Lambda(X)=\{x\in X:X\text{ is 1-dimensional at }x\}$$ is zero-dimensional (i.e. the space $\Lambda(X)$ has a basis of ...
1 vote
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### A special configuration of Nine Circles Theorem and Eight Circle Theorem

The result as follows from special configuration of merge Nine Circle Theorem and Eight Circle theorem but it is new: Problem: Let three circle $(A)$, $(B)$, $(C)$ , let $A_c$ be arbitrary point in ...
1 vote
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### Tiling the plane with pair-wise non-congruent and mutually similar triangles

Question: Is it possible to tile the plane with triangles that are (1) mutually similar, (2) pairwise non-congruent and (3)non-right? No other constraints. Note 1: Reg requirement 3 above: since any ...
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### On partitioning convex planar regions into congruent pieces - 2

We add a bit to A claim on partitioning a convex planar region into congruent pieces . Definition: A perfect congruent partition of a planar region $C$ is a partition of it into some finite number $n$ ...
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### Is there an L-system for aperiodic tilings of the plane with the "hat" monotile?

Most aperiodic tilings of the plane, except possibly for spiral tilings like the Voderberg tiling, exhibit a fractal pattern of self-similarity. This is no exception for the recently discovered "...
327 views

### On the aperiodic monotile

One of the more mind-boggling aspects of the Penrose tiles is that there are uncountably many distinct tilings of the plane, but every tiling contains every finite region that appears in another ...
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### Concurrencies determined by intersections of angle trisectors (and isogonal lines) in a triangle

The famous Morley’s theorem, states that in a triangle the interior angle trisectors, proximal to sides respectively, meet at the vertices of an equilateral. However the six trisectors meet at 12 ...
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### Aperiodic monotile without reflections?

The recently discovered amazing aperiodic monotile (or "einstein") of David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss tiles the plane only if reflections of the ...
719 views

### How can one construct a four-coloring of a tiling of the plane with Smith, Myers, Kaplan, and Goodman-Strauss's aperiodic monotile?

This is motivated by the new paper of Smith, Myers, Kaplan, and Goodman-Strauss, wherein they define the existence of an aperiodic monotile. Clearly their tiling is not three-colorable, so we have ...
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### Is Morley’s observation complete?

Morley’s observation states that in a triangle the intersections of trisectors proximal to a (triangle) side lie six by six on three triples of parallel lines that make angles of 60° with each other. ...
1 vote
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### Reconstructing an ellipse from an arc, synthetically

Given only an arc of a circle, we can easily reconstruct it fully without any use of analytic geometry - indeed using only compass and straightedge. Note that by "given only an arc", we mean ...
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Below, a "disk" means a compact subspace $D \subset \mathbb R^2$ whose boundary is a smooth simple closed curve. Task: Find a procedure which takes as input a pairs of disks $D_0 \subseteq ... 0 votes 0 answers 42 views ### On tiling the plane with non-congruent, equal area triangles with each edge having a unique length Ref: Tiling with incommensurate triangles shows an approach for tiling with incommensurate triangles - all sides and all angles unique and also with different areas - with the perimeters of the tiles ... 1 vote 0 answers 67 views ### Show that a region in a plane defined by a polynomial contains integer points Let$F \in \mathbb{Z}[x,y]$be a polynomial of degree$2n$such that the homogeneous degree$2n$part of$F$, say$F_{2n}$, is positive semi-definite. How does one show that for some$\delta_n > 0$... 4 votes 0 answers 161 views ### The closest ellipse to a given triangle Definition: The Hausdorff distance between two point sets is the greatest of all the distances from a point in one set to the closest point in the other set. Question: Given a general triangle T, to ... 0 votes 1 answer 46 views ### On 'axiality' of planar convex regions Axiality has been studied under a definition given here: https://en.wikipedia.org/wiki/Axiality_(geometry) Consider an alternative definition of axiality as follows: For a convex region C, consider a ... 4 votes 0 answers 96 views ### Do cut-length-minimizing equidissections exist? Suppose$A,B$are polygons of equal area. By the Wallace-Bolyai-Gerwien theorem,$A$and$B$are equidissectable: we can make finitely many straight-line cuts in$A$and rearrange the resulting pieces ... 16 votes 0 answers 377 views ### Is "Escherian metamorphosis" always possible?$\DeclareMathOperator\int{int}\DeclareMathOperator\diam{diam}\DeclareMathOperator\area{area}\DeclareMathOperator\cl{cl}\DeclareMathOperator\ran{ran}\DeclareMathOperator\dom{dom}$This is a tweaked ... 3 votes 1 answer 124 views ### Minimize total area bounded by$N$lines in general position Suppose we have$N$lines in general position (any two lines, but no three lines, meet at a point) ($N\geq 3$). Let the smallest bounded region have area$1$. Determine the minimum (or possibly ... 2 votes 0 answers 80 views ### Ellipse of least perimeter that contains a given triangle This post is related to Smallest 3-ellipses that contain triangles and tries to clarify a basic issue. Question: Given a general triangle T, How does one find and characterize the ellipse of least ... 7 votes 0 answers 88 views ### How many equilaterals have vertices intersections of angle trisectors of a triangle? The celebrated Morley’s theorem ensures that the interior trisectors, proximal to sides respectively, meet at vertices of an equilateral. In the paper Trisectors like Bisectors with Equilaterals ... 2 votes 0 answers 221 views ### Generalization of the Napoleon equilateral triangle to higher dimention When I researched the Fermat-Dao-Nhi equilateral triangle in preamble before points X(33602) of the Kimberling triangle center. I discovered the general result for polygon as follows: Let$A_1$,$A_2$... 3 votes 1 answer 85 views ### Convex polygon shadows: Shortest equivalent segments Let$P$be a convex polygon. Q1. What is the shortest collection of line segments$S$inside$P$with the property that both$P$and$S$have the same sequence of orthogonal shadows as$P$and$S$... 16 votes 1 answer 841 views ### Kakeya crossed-needles problem The Kakeya needle problem asks for the minimum area planar region in which one can completely turn around a line segment through a series of translations and rotations. There is no minimum: There are &... 1 vote 1 answer 87 views ### Triangles that can be cut into mutually congruent and non-convex polygons It is easy to note that an equilateral triangle can be cut into 3 mutually congruent and non-convex polygons (replace the 3 lines meeting at centroid and separating out the 3 congruent quadrilaterals ... 9 votes 1 answer 518 views ### Formula for "cointersection" of three circles? I am working on the problem of finding "rational" dodecahedrons, and I have run across an interesting subproblem: How do you tell if three circles have a common intersection point? ... 19 votes 1 answer 799 views ### All saddles in the unit ball have area$<2\pi$? Let$M$be the saddle surface in$\mathbb R^3$defined by$x^2-y^2+z=0$. For any$r\geq 0$and$(x_0,y_0,z_0)\in\mathbb R^3$, let$rM+(x_0,y_0,z_0)$denotes the surface obtained by scaling$M$by$r$... 2 votes 1 answer 138 views ### Concyclic point made from Six arbitrary points Let$A_1A_2A_3A_4A_5$be irregular convex Pentagon and Let$P$be arbitrary point anywhere in Plane geometry. Let$X_1,X_2,X_3,X_4,X_5$be Circumcircle of$\triangle PA1A3$;$\triangle PA2A4$;$\... 5k views

### Which theorems have Pythagoras' Theorem as a special case?

Loomis famously wrote hundreds of proofs of Pythagoras' Theorem (reference below), but these are all basically proofs "from below". Today on Twitter @panlepan mentioned Carnot's theorem ...
1 vote
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### Fermat point amidst polygonal obstacles

Consider $k$ distinct points in 2D-plane with $n$ convex polygonal obstacles. Is there a poly-time algorithm (poly in $k$ and the total number of obstacle vertices) to find a point outside of all ...
1 vote
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### Writing the plane as {(x,y,z): x+y+z=0} [closed]

One can coordinatize the plane by choosing three axes at 120 degree angles and representing points by triples $(x,y,z)$ with $x+y+z=0$. Is there an accepted name for this kind of coordinate system? (...
1 vote
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### Number of orbits for abelian group actions

Suppose $G$ is an abelian group acting faithfully on two sets, $X$ and $Y$, of the same size. None of $G$, $X$ and $Y$ is finite. Now suppose $G$ is the union of abelian groups $G_i$, where $i$ varies ...
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### A generalization of the law of tangents

The law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. Let $a$, $b$, and $c$ be the lengths of the three ...
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### A plane ray which limits onto itself

A ray is a continuous one-to-one image of the half-line $[0,\infty)$. If $f:[0,\infty)\to \mathbb R ^2$ is continuous and one-to-one, then we say that the ray $X=f[0,\infty)$ limits onto itself if for ...
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### How big can a triangle be, whose sides are the perpendiculars to the sides of a triangle from the vertices of its Morley triangle?

Given any triangle $\varDelta$, the perpendiculars from the vertices of its (primary) Morley triangle to their respective (nearest) side of $\varDelta$ intersect in a triangle $\varDelta'$, which is ...
1 vote
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### On convex planar regions that can be cut into only a specified number of mutually congruent and connected pieces

References: https://math.stackexchange.com/questions/1838617/dividing-an-equilateral-triangle-into-n-equal-possibly-non-connected-parts On congruent partitions of planar regions https://research....
1 vote
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### Minimum bounding rectangle of symmetric convex bodies in the plane : is the ball the worst case

The minimal rectangle containing the euclidean ball in the plane is the standard cube $B_\infty = [-1;1]^2$. I would like to know if the euclidean ball is the worst symmetric convex body to be ...
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### Trigonometry and plane geometry

This will be a variation on the theme of this question, or maybe a rephrasing of it with a somewhat readjusted emphasis. In this posting I introduced the function \begin{align} & f_3(\theta_1,\...
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### An open triangle problem in plane geometry

Some years ago, I asked some 'famous' people in an advanced Plane Geometry forum about the following: Let $ABC$ be arbitrary triangle, how can one construct a point $P$ in the plane such that $P$ is ...
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### Two triangles have the same centroid theorem

Let $\triangle ABC$ and $\triangle A'B'C'$ be two triangles. The line through $A$ and perpendicular to $AA'$ meets the line through $B'$ and perpendicular to $BB'$ at $A_b$; The line through $A$ and ...
1 vote
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### Four incenters lie on a circle-Does this theorem have a name?

When I read the new paper 100 CHARACTERIZATIONS OF TANGENTIAL QUADRILATERALS-section 7, I remember that I posed some problem associated with tangential quadrilateral from 2014 in here and 2015 in here ...
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### Are there infinitely many "generalized triangle vertices"?

Briefly, I'd like to know whether there are infinitely many "generalized triangle centers" which - like the orthocenter - are indistinguishable from a vertex of the original triangle. This ...
210 views

### Least area and least perimeter triangles that contain a convex planar region - how different can they be?

Is there a planar convex region whose enclosing triangles of least perimeter and least area have different areas and different perimeters? And if so, which region maximizes the difference between the ...
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### What is the average component size of a coloring?

Supose each cell of a big (or infinite) grid is colored at random by one of $k$ colors. Then the connected monochromatic components (here components are not supposed to contain "wasp waists",...
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### Maximizing the area of a region involving triangles

I thought of a question while making up an exercise sheet for high school students, and posted it on MathStackExchange but did not receive an answer (the original post is here), so I thought perhaps ...
1 vote
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### To extend the Steiner-Lehmus theorem

The Steiner Lehmus theorem (https://en.wikipedia.org/wiki/Steiner%E2%80%93Lehmus_theorem) states: Every triangle with two angle bisectors of equal lengths is isosceles. Question: What could one say ...
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### Is a line associated with antipodal points (the fact, it is the generalization of Simson line) known?

First time, I found a line associated with antipodal points, detail: Let $ABC$ be a triangle, $(C)$ is circumconic of $ABC$. $P$ and $P'$ are two antipodal points. Construct three lines through $P'$ ...
1 vote
### Lines passing through many points of the form $(c^n,c^m)$
For $c>1$ consider the subset $X\subset \mathbb R^2$ consisting of all points $(c^n,c^m)$ where $n,m\in \mathbb Z$. Question. Suppose $L\subset \mathbb R^2$ is a line that is not horizontal, not ...